Breathing From ACC Formula Sheet

Use this sheet when you want the calibration, projection, bound selection, and volume mapping behind the Breathing tab written out clearly, with an explicit validation disclaimer.

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Provenance

This breathing-volume model is repository-specific code ported from the existing Unity runtime. It is not presented as a published external method reference.

Important status note:

this ACC breathing-volume model has not yet been externally validated
a validation study is actively being worked on

Data Path

input stream: PMD accelerometer samples from the Polar H10
preprocessing: exponential moving average on the raw ACC signal
calibration: estimate a center point and a principal breathing axis from a calibration window
projection: map filtered ACC onto the chosen breathing axis
bounds: derive lower and upper projection bounds from calibration quantiles
output: convert the projected signal into a 0..1 breathing-volume estimate

In the notation below, a_raw(t) is the raw accelerometer vector at time t, a_f(t) is the filtered vector, alpha is the smoothing factor, c is the calibration center, u is the dominant breathing axis, p(t) is the scalar projection onto that axis, and Q_lower / Q_upper are the calibration quantile operators used to define a practical operating range. The tracker works in accelerometer coordinates internally and only turns the signal into a single breathing scalar after the projection step.

Core Approximation

Filtered ACC sample:

a_f(t) = (1 - \alpha)a_f(t-1) + \alpha a_{raw}(t)

This smooths the raw accelerometer stream so the later breathing estimate is less sensitive to frame-level jitter.

Calibration center:

c = \frac{1}{N}\sum_{i=1}^{N} a_f(i)

This is the average filtered posture during calibration and acts as the reference point for later motion.

Principal-axis projection:

p(t) = \langle a_f(t) - c, u \rangle

Where u is the dominant calibration axis estimated from the calibration-window covariance.

This projection turns 3-D chest motion into one signed scalar that should rise and fall mainly along the learned breathing direction.

If the optional XZ model is available, the tracker also keeps a 2-D projection in the chest plane:

p_{xz}(t) = \langle (x(t), z(t)) - c_{xz}, u_{xz} \rangle

This is the same idea as p(t), but constrained to the chest plane for the optional XZ model.

Lower and upper projection bounds come from calibration quantiles:

b_{min} = Q_{lower}(p), \qquad b_{max} = Q_{upper}(p)

These bounds define the expected low and high ends of the breathing excursion without trusting a single noisy minimum or maximum sample.

The runtime then applies the configured edge-ease and optional adaptive-bounds updates before mapping the projection to volume:

v(t) = \operatorname{clamp}\left(\frac{p(t) - b_{min}}{b_{max} - b_{min}}, 0, 1\right)

This maps the projected motion into a normalized 0..1 breathing volume, where values near 0 sit near the lower calibration bound and values near 1 sit near the upper calibration bound.

The final base volume uses either the 3-D axis model or the XZ model, depending on the tracker settings. The displayed output may then be inverted if the configured inhale/exhale direction is flipped.

State Classification

The tracker derives Inhaling, Exhaling, and Pausing from the first difference of the output volume:

\Delta v(t) = v(t) - v(t-1)

Positive Delta v means the estimated volume is rising, negative Delta v means it is falling, and values near zero mean the tracker sees little movement in either direction.

Inhaling when Delta v exceeds the configured positive threshold
Exhaling when Delta v is below the negative threshold
Pausing otherwise

Alignment Status

There is no external paper claim here. The correct interpretation is:

this page documents what the app computes
the model is intended as a practical ACC-only breathing approximation
external validation is still pending